Comparison of h-, p- and hp-adaptation for convective heat transfer

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1 Computatonal Methods and Expermental Measurements XIII 495 Comparson of h-, p- and hp-adaptaton for convectve heat transfer D. W. Pepper & X. Wang Nevada Center for Advanced Computatonal Methods, Unversty of Nevada Las Vegas, USA Abstract Three adaptve FEM algorthms based on mesh refnement (h-adaptaton), mesh enrchment (p-adaptaton) and the combnaton of both (hp-adaptaton) are employed to solve ncompressble flud flow problems ncludng convectve heat transfer effects. Test cases of natural convecton n a square cavty wth dfferent Raylegh numbers are solved usng prmtve varables n a modfed fnte element approach employng the three adaptve strateges. Results show excellent agreement among benchmark data avalable n the lterature. Keywords: h-, p-, hp- adaptaton, FEM, natural convecton. 1 Introducton The fnte element method (FEM) s a popular numercal tool used n many heat transfer and flud flow smulatons. The FEM s capable of easly dealng wth rregular geometres and has the ablty to mplement enhanced accuracy usng general-purpose algorthms. Adaptve FEM s especally attractve snce t can dynamcally control mesh characterstcs to obtan desred accuracy. Followng early work usng h-adaptve FEM to accurately capture shock waves n compressble flow [1], the adaptve FEM has become an actve research area over the past decade. Generally, there are four categores of adaptaton: (1) h-adaptaton, where the element szes vary whle the order of the shape functons are constant; (2) p- adaptaton, where the order of the shape functons vary whle the element szes are constant; (3) r-adaptaton, where the nodes are redstrbuted n an exstng mesh n the process of adaptaton whle the total element and node number are do: /cmem070501

2 496 Computatonal Methods and Expermental Measurements XIII constant; (4) hp-adaptaton, whch s a combnaton of both h- and p- method. hp-adaptve schemes are among the best mesh based schemes wth the potental payoff of achevng exponental converge rate [2, 3]. In ths paper, three adaptve FEM algorthms based on mesh refnement (hadaptaton), mesh enrchment (p-adaptaton) and the combnaton of both (hpadaptaton) are employed to solve ncompressble flud flow problems wth convectve heat transfer. Results are compared for dfferent algorthms along wth a bref dscusson of computatonal effcency. 2 Governng equatons and soluton procedure The non-dmensonal governng equatons for ncompressble lamnar vscous flud n a Boussnesq and constant property are wrtten n the forms: Contnuty equaton: V = 0 (1) Conservaton of Momentum V 2 + V V = p + Pr V + C T grav (2) t for the x-drecton, C grav = 0 ; for the y-drecton, Cgrav = RaPr. The energy equaton can be wrtten as Energy equaton T 1 + V = 2 T T + Q (3) t Pe The followng non-dmensonal parameters are used to derve the above equatons. * * * * * X V p t T Tc X =, V =, p =, t =, T = (4) L α / L ρα / L L / α Th Tc wth the Raylegh number, Prandtl number, and Peclet number defned as gβ( T ) 3 h Tc L ν Ra =, Pr =, Pe = Re Pr (5) αν α where β s the thermal expanson coeffcent, α s the thermal dffusvty, ν s the knematc vscosty, T h and Tc are for hot and cold wall temperature respectvely. A proecton method, also known as a fractonal step method, s used for the flow solver. Ths method s based on the Helmholtz-Hodge Decomposton Theorem (Chorn [4]), and detaled descrpton of employment of proecton method can be found n the work of Ramaswamy et al. [5]. 3 Fnte element formulatons Quadrlateral elements are used to dscretze the problem domans. The Galerkn weghted resdual method s used. The varables V and T can be replaced by usng the tral functons:

3 Computatonal Methods and Expermental Measurements XIII 497 n = 1 ( ) ( ) V( x, t) = N x V t (6) n = 1 ( ) ( ) T( x, t) = N x T t (7) where x s the computatonal doman, s the degree of freedom (DOF) ndex and n s the number of DOFs. Under proecton algorthms, the weghted resdual forms of the momentum and energy equatons can be wrtten as (summaton conventon s mpled) Momentum: NNd V + N ( NV k k) d { V} + Pr d { V } x x x (8) f ( x) Nd CvscN { V} nd Γ = 0 x where denotes (x) and Γ represents the boundares of the computatonal doman. For the vertcal component (y) wth natural convectve effects: f x = C T = RaPr T ( ) { } { } grav Energy: x x x NNd T + N ( NV k k) d { T} + d { T} QN d N qγdγ = 0 Equatons (8) and (9) can be wrtten n matrx form as [M]{V} + ([K ] + [A(V)]){V} = {F } (10) v [M]{T} + ([K T] + [A(V)]){T} = {F T} (11) where the over dot refers to tme dfferentaton. The matrx coeffcents are defned as: [ M ] = NNd (12) A V = N N V d ( ) ( ) V (9) k k (13) x [ K ] Pr V [ K ] T = d (14) x x = d (15) x x V { Fv} = Nf ( x) d+ Pr Nn dγ (16) Γ x

4 498 Computatonal Methods and Expermental Measurements XIII { } (17) F = NQd+ NqdΓ T Γ A Petrov-Galerkn scheme s employed to weght the advecton terms n the governng equatons. The altered weghtng functon skews the nterpolaton functon n the upwnd drecton so that the dsperson and added dffuson ntroduced by the standard Galerkn formulaton are mnmzed,.e. ˆ αhe W = N + V N 2 V (18) γ 2 ˆ α = coth 2 γ (19) where ˆα s the Petrov-Galerkn weghtng factor, h e s the characterstc element length and γ s the Petrov-Galerkn stablty parameter. For flow wth natural convecton, γ = VhRa e Pr. Mass lumpng s used n order to obtan a fully explct tme marchng scheme,.e. M = m (20) 4 Adaptaton methodology 4.1 Error estmator [ ] 1 1/ Varous error estmators exst that can be used n adaptaton, e.g., the element resdual method, nterpolaton methods, subdoman-resdual methods, and proecton method. The rght chosen error estmator s the bass for a successful adaptaton procedure. Detaled descrptons of dfferent error estmators can be found n [6 9]. In ths study, an error estmator was chosen based on an extenson of the work by Zenkewcz and Zhu [7] due to ts reasonable accuracy, smplcty and ease of mplementaton. The errors n a fnte element soluton are the dfference between the exact and approxmate solutons, whch can be expressed n certan norms such as the Energy norm or L 2 norm. In ths smulaton, the L 2 norm s adopted. The correspondng stress error measure can be wrtten as 1/2 T eσ = eσeσd (21) and all element errors are typcally defned as: 2 σ e m = 1 2 = e (22) where m stands for the total number of elements. The error ndces η = η n the form of error percentage defned as: σ σ 2 ( ) 1/2 * σ e 2 2 ησ = eσ / + σ 100% (23)

5 Computatonal Methods and Expermental Measurements XIII 499 The error ndex η s used to gude the adaptaton procedure. Temperature s set for the key adaptaton varable n our paper. 4.2 Adaptaton rules Unstructured meshes, ansotropc and 1-rregular mesh adaptaton rule for h-adaptaton An unstructured, ansotropc mesh s allowed whch s an effcent, drectonal refned mesh where refnement n one drectonal s needed. The 1-Irregular mesh refnement rule allows an element to be refned only f ts neghbors are at the same or hgher level (1-Irregular mesh). By followng ths rule, multple constraned nodes (parent node themselves are constrant nodes) can be avoded Mnmum rule for p-adaptaton Herarchcal shape functons employed n p-adaptaton can be categorzed as: node functons, edge functons, face functons (for 3D cases) and bubble functons. The mnmum rule states that the order for an edge common for two elements never exceeds orders of the neghborng mddle nodes. For quadrlateral elements n 2D, both the horzontal and vertcal orders must be consdered hp constrants are employed to meet contnuty requrements As a combnaton of h- and p- adaptaton, hp-adaptaton can be ether refned (unrefned) or enrched (unenrched) whenever necessary. The adaptaton rules for h- and p- are combned n hp-adaptaton. In addton, to mantan contnuty of global bass functon, constrants at the nterface of elements supportng edge functons of dfferent order are employed. The constrant represents a generalzaton of the hp-constrants, whch s dscussed n Demkowcz et al [10]. 4.3 Adaptaton strategy An acceptable soluton s reached when global and local error condtons are met. A global error condton states that the global percentage error should not be greater than a maxmum specfed percentage error: η ηmax. If η > ηmax, a new teraton s performed. The local error condton states that local relatve percentage error of any sngle element e σ should not be greater than the averaged error e avg among all the elements n the doman. The average element error s defned as: 1/2 * 2 2 eavg = η max ( σ + eσ ) / m (24) A local element refnement ndcator ξ = e /e avg s defned to decde f a local refnement for an element s needed: when ξ > 1, the element s refned; when ξ < 1the element s unrefned. In an h-adaptve process, the new element sze s calculated usng: 1/p h = h / ξ (25) new old

6 500 Computatonal Methods and Expermental Measurements XIII In a p-adaptve process, the new shape functon order s calculated usng: 1/ p pnew = poldξ (26) The hp-adaptve strategy used n ths study employs a L 2 norm error estmator, whch s an extenson from the three-step hp-adaptve strategy developed by Oden et al. [11]. In the hp-adaptve procedure, a sequence of refnement steps s employed. Three consecutve hp-adaptve meshes are constructed for solvng the system n order to reach a preset target error: ntal mesh, the ntermedate h-adaptve mesh, and the fnal hp- adaptve mesh obtaned by applyng p-adaptve enrchments on the ntermedate mesh. The p- adaptaton s carred out when the problem soluton s pre-asymptotc. 5 Numercal results 5.1 Problem defnton Natural convecton n a dfferentally heated cavty has been studed extensvely for over 30 years (dfferentally heated vertcal walls - hot left and cold rght; adabatc horzontal walls top and bottom). Numercal smulaton results are generally compared wth the accurate benchmark solutons obtaned by De Vahl Davs [12], who used a FDM wth a stream functon - vortcty formulaton. In ths study, the problem was solved usng h-, p- and hp-adaptve FEM wth Pr = 0.71, Ra = Smulaton results Results were obtaned for Ra = Excellent agreement was observed over the range of Ra numbers wth data avalable n lteratures [12]. Results from the lower Ra number computatons are essentally duplcatve wth results found n the lterature. Results for Ra =10 5 and 10 6 are presented. Fnal adapted meshes and results for h-adaptaton are shown n Fgure 1. 3-level h-adaptaton was employed and produced 2503 elements wth 2542 DOFs for Ra=10 5. Fnal adapted meshes and results for p-adaptaton are shown n Fgure 2. Only 3-level p-adaptaton was used and created 400 elements wth 1184 DOFs for Ra=10 5. (a) (b) (c) Fgure 1: h-adaptve results for Ra=10 5 (a) fnal adapted mesh, (b) streamlnes, and (c) temperature contours.

7 Computatonal Methods and Expermental Measurements XIII 501 (a) (b) (c) Fgure 2: p-adaptve results for Ra=10 5 (a) fnal adapted mesh, (b) streamlnes, and (c) temperature contours. (a) (b) (c) Fgure 3: hp-adaptve results for Ra=10 5 (a) fnal adapted mesh, (b) streamlnes, and (c) temperature contours. (a) (b) (c) Fgure 4: hp-adaptve results for Ra=10 6 (a) fnal adapted mesh, (b) streamlnes, and (c) temperature contours. Fnal adapted meshes and results for hp-adaptaton are shown n Fgures 3 and 5. In these smulatons, both 3-level h- and p-adaptatons were used resultng n 1368 elements and 6897 DOFs for Ra=10 5. The test case for Ra=10 6 produced a fnal mesh consstng of 1372 elements and 6529 DOFs, whch s shown n Fgure 4.

8 502 Computatonal Methods and Expermental Measurements XIII All the adaptve results agree well wth benchmark data avalable n the lterature for both flow and thermal patterns [12]. Quanttatve studes were also conducted for the hp-adaptve algorthm wth Ra=10 6. Comparsons wth [12] were made for the maxmum horzontal and vertcal veloctes together wth ther locatons on the vertcal and horzontal mdplane; Nu 0, the average Nusselt number on the heated wall; and the maxmum and mnmum values of local Nusselt number on the heated sde together wth ther locatons. Results of the comparson study are shown n Table 1. 6 Dscusson The adaptve algorthms change element sze and shape functon order dynamcally. The adaptatons are based on key varable gradents of the error dstrbuton, subsequently producng hghly accurate values wth a mnmum of computatonal cost. All three adaptve schemes are effcent n reducng overall computatonal tme. Ths s partcularly true for h-adaptaton compared wth a global unformly refned algorthm. The p-adaptve algorthm s also effectve when compared wth a unformly enrched algorthm. The hp-adaptve algorthm s the best of the three adaptve schemes, and has been shown to be exponentally convergent compared wth unformly refned and enrched technques [13]. Table 1: Comparson of hp-fem results wth benchmark data (Ra=10 6 ). [12] results hp-fem results u max y (x=0.5) v max x (y=0.5) Nu Nu max y (x=0) Nu mn y (x=0) 1 1 A globally unform h-refned and p-enrched mesh (unformly refned up to 3- levels and enrched up to 3 rd order) and the hp-adaptve algorthm (adaptvely refned up to 3-levels and enrched up to 3 rd order) were analyzed for Ra = The ntal mesh for the hp-adaptve algorthm was 10 x 10. Results showed that usng a globally h-refned and a p-enrched algorthm consumed nearly 18X more CPU tme (proected) than the hp- adaptve algorthm, as shown n Table 2. The choce of usng a partcularly type of adaptve algorthm and selecton of key varables depends on varous problem constrants and propertes desred by the user. For example, a smple rectangular doman wth smple boundary condton constrants can be solved wthout adaptaton usng conventonal numercal methods. On the other hand, complex geometres and regons where

9 Computatonal Methods and Expermental Measurements XIII 503 hgh gradents can occur are best handled usng dynamc adaptve technques and elmnate the burden on the user of havng to remesh the problem. A lst of the advantages and dsadvantages for each adaptve scheme s shown n Table 3. Table 2: CPU tme comparson between unform refned and enrched algorthm and hp-adaptve algorthm. Compare Cases # of element Intal Fnal # of DOF Intal Fnal Total CPU Tme Per DOF CPU tme # of teraton Unform h and p hpadaptve algorthm ,884 (sec) (sec) 9.62 (sec/ DOF) 5.59 (sec/ DOF) Table 3: Comparson of h-, p-, r- and hp- adaptve algorthm. h- adaptaton p- adaptaton r- adaptaton hpadaptaton element sze varous constant varous varous DOF (degrees varous varous constant varous of freedom) shape functon constant varous constant varous advantages elements wll not become overly dstorted relatve coarse mesh may be suffcent no new nodes added exponental convergence rate dsadvantages 7 Conclusons dffculty n dealng wth constrant nodes codng complexty elements may become overly dstorted dffculty n dealng wth constrant nodes and codng complexty Three adaptve algorthms now beng used n the fnte element method have been developed to solve for flud flow and heat transfer. Natural convecton wthn a dfferentally heated enclosure was solved usng h-, p- and hp-adaptve algorthms. Smlar flow and thermal patterns were observed n all three adaptve solutons for 10 3 Ra Excellent agreement was obtaned for all three methods compared wth benchmark data avalable n the lterature. Characterstcs for the dfferent adaptve algorthms are dscussed. The

10 504 Computatonal Methods and Expermental Measurements XIII computatonal effcency for the hp-adaptve algorthm showed an 18X decrease n CPU tme compared to usng a unformly refned and enrched FEM algorthm. Adaptve algorthms are especally promsng n dealng wth problems where soluton and error dstrbutons are hard to predct. References [1] Perare J., Vahdat M., Morgan K., Zenkewcz O.C., Adaptve Remeshng for Compressble Flows, J. Comput. Physcs, 72 (2), pp26-37, 1987 [2] Guo, B. and Babuska, I., The h-p Verson of the Fnte Element Method, Parts 1 and 2, Comp. Mech., Vol. 1, pp and pp , [3] Gu, W. and Babuska, I., The h, p and h-p Verson of the Fnte Element Method n One Dmenson, Parts 1 and 2, Numersche Mathematk Vol. 49, pp , [4] Chorn, A. J., Numercal Soluton of the Naver-Stokes Equatons, Math. Comp., Vol. 22, pp , [5] Ramaswamy, B., Je, T. C. and Akn, J. E., Sem-Implct and Explct Fnte Element Schemes for Coupled Flud/Thermal Problems, Int. J. Num. Meth. Engng., Vol. 34, pp , [6] P. Ntharasu and O. C. Zenkewcz, Adaptve mesh generaton for flud mechancs problems, Int. J. Num. Meth. Engng. 47, pp , [7] Zenkewcz O. C. and Zhu R. J. Z., A Smple Error Estmator and Adaptve Procedure for Practcal Engneerng Analyss, Int. J. Num. Meth. Engng., Vol. 24, pp , [8] Oden, J. T., Demkowcz, L. Rachowcz, W. and Westermann T. A., Toward a Unversal h-p Adaptve Fnte Element Strategy, Part 2. A Posteror Error Estmaton, Comp. Meth. Appl. Mech. and Engng., Vol. 77, pp , [9] Answorth, M. and Oden, J. T., A Posteror Error Estmaton n Fnte Element Analyss, Pure and Appled Mathematcs, A Wley-Interscence Seres of Texts, Monographs and Tracts, [10] Demkowcz, L, Oden, J. T., Rachowcz, W and Hardy, O, Toward a Unversal h-p Adaptve Fnte Element Strategy, Part 1. Constraned Approxmaton and Data Structures, Comp. Meth. Appl. Mech. and Engng, Vol. 77, pp , [11] Oden, J. T., Wu, W. and Answorth, W., Three-Step h-p Adaptve Strategy for the Incompressble Naver-Stokes Equatons, Modelng, Mesh Generaton, and Adaptve Numercal Methods for Partal Dfferental Equatons, Sprnger-Verlag, pp , [12] De Vahl Davs, G., Natural Convecton of Ar n a Square Cavty: A Bench Mark Numercal Soluton, Int. J. Numercal Methods n Fluds, Vol. 3, , [13] Wang, X. and Pepper, D. W., Applcaton of an hp-adaptve technque for heat, mass and momentum transport, ASME Int. Mech. Engng. Cong. and Exp., Nov.5-11, 2005 Orlando, Florda.

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